Adaptive density estimation based on a mixture of Gammas

Abstract

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local H\"older densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness

Nonparametric Bayesian Deconvolution of a Symmetric Unimodal Density

“…This is achieved by a mixture model with a mixing distribution modelled by a Dirichlet Process mixture of Gamma distributions. We obtain some preliminary results on this layer from Bochkina and Rousseau (2017). An inversion inequality is derived that bridges our theory from p(•) to f (•).…”

“…using a Dirichlet process location-mixture of Gamma distributions, which has large support (Bochkina and Rousseau, 2017) on densities supported on R + , and is easy to implement in a Bayesian framework. Specifically, we reparameterize a Gamma density by its shape z and mean µ as parameter pairs.…”

“…For notational simplicity, we drop the arguments and only use M to denote the space of densities in Condition 1. Similar function spaces with additional smoothness assumptions have been used by Bochkina and Rousseau (2017); we do not make such smoothness assumptions here. The conditions are typical in the literature on Bayesian density estimation.…”

“…Condition 3 mainly controls the prior thickness of the sieve space upon which the inversion inequality in Section 3.3 can be derived. Bochkina and Rousseau (2017) showed…”

“…We provide large-sample theoretical support to the proposed methodology by showing posterior consistency for the observed density and the latent density. For the observed density of W , we borrow results from recent work (Bochkina and Rousseau, 2017) where posterior convergence rates for estimating a density on the positive half-line were established using Dirichlet location-mixtures of Gamma distributions. Their setup nicely serves as a component in our hierarchical model for the density of W .…”